Geometric Sequences and Series

A sequence is geometric if the ratios of consecutive terms are the same.

2, 8, 32, 128, 512, . . .

geometric sequence

The common ratio, r, is 4.

82

4

328

4

12832

4

512128

4

Example 1.

a. Is the sequence geometric? If so, what is ?

2,4,8,16,...2 ,...n

r

4 8 162, 2, 2

2 4 8 2r

b. Is the sequence geometric? If so, what is ?

1 1 1 1 1 , , , ,..., ,...

3 9 27 81 3

n

r

1 1 11 1 19 27 81

1 1 13 3 33 9 27

1

3r

The nth term of a geometric sequence has the form

an = a1rn - 1

where r is the common ratio of consecutive terms of the sequence.

The nth Term of a Geometric Sequence

15, 75, 375, 1875, . . . a1 = 15

The nth term is 15(5n-1).

75 515

r

a2 = 15(5)

a3 = 15(52)

a4 = 15(53)

1Example 2. Write the first five terms of the geometric sequence whose first term is a 3

and whose common ratio is 2.r

1

12

3

3 2 6

a

a

2

3 3 2 12a 3

4 3 2 24a 4

5 3 2 48a

Example 3. Find the 15th term of the geometric sequence whose first term is 20 and whose common ration is 1.05.

11n

na a r

14

15 20 1.05a 39.599

Example 4. Find a formula for the nth term of the following geometric sequence. What is the ninth term of the sequence?

5, 15, 45, …

Find the common ratio

15 5 3 45 15 3

15 3

n

na

8

9 5 3a 32,805

125Example 5. The 4th term of a geometric sequence is 125, and the 10th term is .

64Find the 14th term. Assume that the terms of the sequence are positive.

10 410 4a a r

6125125

64r

61

64r

1

2r

414 10a a r

4125 1

64 2

125

1024

The sum of the first n terms of a sequence is represented by summation notation.

Definition of Summation Notation

1 2 3 41

n

i ni

a a a a a a

index of summation

upper limit of summation

lower limit of summation

5

1

4n

n

1 2 3 4 54 4 4 4 4 4 16 64 256 1024 1364

The Sum of a Finite Geometric Sequence

The sum of a finite geometric sequence is given by

11 1

1

1 .1

n nin

i

rS a r ar

5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?

n = 8

a1 = 5

1

81 11

221

5n

nrS ar

5210r

1 25651 2 2555

1 1275

12

1

Example 6. Find the sum 4 0.3n

n

Write out a few terms.

12

1 2 3 12

1

4 0.3 4 0.3 4 0.3 4 0.3 ... 4 0.3n

n

1 4 0.3 0.3 and 12a r n

12

11

14 0.3

1

nn

n

ra

r

121 0.3

4 0.31 0.3

1.714

If the index began at i = 0, you would have to adjust your formula

12 12

0

0 1

4 0.3 4 0.3 4 0.3n

n

i n

12

1

4 4 0.3n

n

4 1.714 5.714

Definition of Geometric Series

The sum of the terms of an infinite geometric sequence is called a geometric series.

a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . .

If |r| < 1, then the infinite geometric series

11

0

.1

i

i

aS a r

r

has the sum

If 1 , then the series does not have a sum.r

Example 7. Use a graphing calculator to find the first six partial sums of the series. Then find the sum of the series.

1cum sum(seq(4*0.6 , ,1,6))x x

4, 6.4, 7.84, 8.704, 9.2224, 9.53344

Use the formula for the sum of an infinite series to find the sum.

410

1 0.6S

1

1

4 0.6n

n

Example 8. Find the sum of 3 + 0.3 + 0.03 + 0.003 + …,

33.33

1 0.1S

Example 9. A deposit of $50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance at the end of 2 years?

This type of savings plan is called an increasing annuity.

The first deposit will gain interest for 24 months, and its balance will be

The second deposit will gain interest for 23 months

24

24

24

0.0650 1 50 1.005

12A

23

23

23

0.0650 1 50 1.005

12A

The last deposit will gain interest for only one month

1

1

0.0650 1 50 1.005

12A

The total balance will be the sum of the balances of the 24 deposits.

24

1

1 1.005150 1.005 $1277.96

1 1 1.005

n

n

rS a

r

Homework

Page 607-608

2-24 even, 25-31 odd, 33, 35, 40, 42, 48-54 even